Calculus

The Computational Method (mathematics)

The Mineral growth in a hollow organ of the body, e.g. kidney stone (medical term)

Function

• A function is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B.

Linear Function

• y = f (x) = mx + b

• where m is the slope of the line and b is the y-intercept.

ENGINEERING EXAMPLE(a) As dry air moves upward, it expands and cools. If the ground temperature is 20 °C and the temperature at a height of 1 km is 10 °C , express the temperature T (in °C) as a function of the height h (in kilometers), assuming that a linear model is appropriate.(b) Draw the graph of the function in part (a). What does the slope represent?(c) What is the temperature at a height of 2.5 km?

SOLUTION

(a) Because we are assuming that T is a linear function of h, we can write

T = mh + b

We are given that T = 20 °C when h = 0, so

20 = m . 0 + b = b

In other words, the y-intercept is b = 20.

We are also given that T = 10 °C when h = 1, so

10 = m . 1 + 20

The slope of the line is therefore m = -10 and the required linear function is

T = -10h + 20

(b) The graph is sketched in Figure 3. The slope is m = -10 °C/km, and this represents the rate of change of temperature with respect to height.

(c) At a height of 2.5 km, the temperature isT = -10(2.5) + 20 = - 5 °C

Student Assignment9. The relationship between the Fahrenheit and Celsius temperature scales is

given by the linear function :

(a) Sketch a graph of this function.(b) What is the slope of the graph and what does it represent?What is the F-intercept and what does it represent?

Polynomials Function• Quadratic function = Polynomial degree 2

P(x) = ax2 + bx + c

• Cubic function = Polynomial degree 3

P(x) = ax3 + bx2 + cx + d

Polynomial n-degree• P(x) = anxn + an-1xn-1 +…+ a2x2 + a1x + a0

EXAMPLE 4 A ball is dropped from the upper observation deck of the CN Tower, 450 mabove the ground, and its height h above the ground is recorded at 1-second intervals in

Table 2. Find a model to fit the data and use the model to predict the time at which theball hits the ground.

SOLUTION

We draw a scatter plot of the data in Figure 9 and observe that a linear model is inappropriate. But it looks as if the data points might lie on a parabola, so we try a quadratic model instead. Using a graphing calculator or computer algebra system (which uses the least squares method), we obtain the following quadratic model:

h = 449.36 + 0.96t + 4.90t 2

In Figure 10 we plot the graph of The Equation together with the data points and see that the quadratic model gives a very good fit. The ball hits the ground when h=0, so we solve the quadratic equation:

ax2 + bx + c = 0,

The quadratic formula gives

The positive root is t 9.67, so we predict that the ball will hit the ground after about 9.7 seconds.

Student Assignment

7.6 Let f (x) = x2 + 2x − 1 for all x. Evaluate:

(a) f (2),

(b) f (−2),

(c) f (−x),

(d) f (x + 1)

(e) f (x − 1)

(f) f (x + h)

(g) f (x + h) − f (x)

(h) f (x + h) − f (x)

h

Power FunctionA function of the form f (x) = xa, where is a constant, is called a power function.

(i) a = n, where n is a positive integer

(ii) a = 1/n, where n is a positive integer. The function is a root function.

(ii) a = -1. The function is a reciprocal function.

Rational Function• A rational function f is a ratio of two polynomials:

Trigonometric Function

Exponential Function• The exponential functions are the functions of the form where the base a is a

positive constant.

Logarithmic Function